Simplify (m^-2 n^-6 p^5) / (m^7 n^4 p^-3) and write the answer using only positive exponents.

Step-by-step solution

1. 1

   Handle one base at a time. The quotient rule $\dfrac{a^{s}}{a^{t}}=a^{s-t}$ applies separately to $m$, $n$ and $p$; there is no need to convert the negative exponents first, and doing so tends to create sign errors.

2. 2

   Subtract exponents for m.

   $\frac{m^{-2}}{m^{7}}=m^{-2-7}=m^{-9}.$

3. 3

   Subtract exponents for n and p. Watch the double negative in the $p$ denominator — subtracting $-3$ adds $3$:

   $\frac{n^{-6}}{n^{4}}=n^{-6-4}=n^{-10},\qquad \frac{p^{5}}{p^{-3}}=p^{5-(-3)}=p^{8}.$

4. 4

   Collect the three results.

   $m^{-9}n^{-10}p^{8}.$

5. 5

   Convert to positive exponents. A negative exponent means the factor belongs on the other side of the fraction bar, with the sign of the exponent flipped:

   $m^{-9}n^{-10}p^{8}=\frac{p^{8}}{m^{9}n^{10}}.$

6. 6

   Check with a quick substitution. At $m=n=p=2$ the original is $\frac{2^{-2}\cdot 2^{-6}\cdot 2^{5}}{2^{7}\cdot 2^{4}\cdot 2^{-3}}=\frac{2^{-3}}{2^{8}}=2^{-11}$, and the answer gives $\frac{2^{8}}{2^{9}\cdot 2^{10}}=2^{8-19}=2^{-11}$. They agree.